Summary & Master Memory Capsule

Master Formula Sheet

#	Result	Formula
1	Mirror focal length	$f = R/2$
2	Mirror equation	$\frac{1}{v}+\frac{1}{u}=\frac{1}{f}$; $m = -v/u$
3	Snell's law	$\frac{\sin i}{\sin r} = n_{21} = \frac{v_1}{v_2}$; $n = c/v$
4	Index algebra	$n_{12} = 1/n_{21}$; $n_{32} = n_{31}n_{12}$
5	Apparent depth	real depth / n (normal viewing); slab raise $t(1-1/n)$
6	Critical angle	$\sin i_c = 1/n$; TIR if denser$\to$rarer and $i > i_c$
7	Spherical surface	$\frac{n_2}{v}-\frac{n_1}{u}=\frac{n_2-n_1}{R}$
8	Lens maker	$\frac{1}{f}=(n_{21}-1)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$
9	Thin lens	$\frac{1}{v}-\frac{1}{u}=\frac{1}{f}$; $m = v/u$
10	Power	$P = 1/f$ (dioptre, f in m); contact: $P = \sum P_i$, $m = \prod m_i$
11	Prism	$r_1+r_2 = A$; $d = i+e-A$; $n = \frac{\sin[(A+D_m)/2]}{\sin(A/2)}$; thin: $D_m=(n-1)A$
12	Magnifier	$m = 1 + D/f$ (near point) or $D/f$ (infinity), D = 25 cm
13	Microscope	$m = \frac{L}{f_o}\cdot\frac{D}{f_e}$ (L = tube length)
14	Telescope	$m = f_o/f_e$; tube $= f_o + f_e$

The Sign Convention & Quick Verdicts

Three lines to rule them all: measure from the pole/optical centre; along incident light = positive; heights up = positive.

Item	Concave mirror	Convex mirror	Convex lens	Concave lens
f	negative	positive	positive	negative
Real object u	negative	negative	negative	negative
Always-image	case map	virtual, erect, diminished	case map	virtual, erect, diminished

Magnification decoders: mirrors $m = -v/u$; lenses $m = +v/u$. In both: m negative = real-and-inverted; m positive = virtual-and-erect.

Case-map pivots (concave mirror / convex lens): object at C/2F → same size; at F → image at infinity; inside F → the magnifier (virtual, erect, enlarged).

TIR applications: 90/180-degree prisms (need $i_c < 45^{\circ}$), image inverters, diamond brilliance ($i_c = 24.4^{\circ}$), optical fibres (denser core, rarer cladding). Mirror f never changes in water; lens f grows (relative index falls); a lens in matched liquid vanishes.

Instruments at a Glance

Feature	Compound microscope	Astronomical telescope
Object	tiny, just outside $f_o$	huge, at infinity
Objective	small $f_o$, small aperture	LARGE $f_o$, LARGE aperture
First image	real, inverted, magnified	real, inverted, at the focus
Eyepiece	magnifier ($D/f_e$ or $1+D/f_e$)	magnifier
Total m	$\frac{L}{f_o}\frac{D}{f_e}$ (250 in NCERT's example)	$\frac{f_o}{f_e}$ (100 in NCERT's)
Tube	L between the foci	$f_o + f_e$
Final image	inverted	inverted (terrestrial adds erecting lenses)

Reflecting telescopes (Cassegrain) win because mirrors have no chromatic aberration, a parabolic figure kills spherical aberration, and mirrors can be supported across the back — enabling the huge apertures that gather light and resolve detail.

One-Glance Revision Flow

The chapter in seven steps:

1. Sign convention first — it powers every formula.
2. Mirrors: $f = R/2$, plus-form equation, $m = -v/u$; convex = always-diminished-erect.
3. Refraction: Snell, index chains, slab shift, apparent depth = real/n.
4. TIR: $\sin i_c = 1/n$; denser-to-rarer AND $i > i_c$; prisms, diamonds, fibres.
5. Surfaces $\to$ lenses: one-surface master formula, applied twice = lens maker's; thin-lens minus-form equation; $P = 1/f$, powers add in contact.
6. Prism: $d = i+e-A$; the $(A, D_m)$ formula measures n; 60-30-$\sqrt2$.
7. Instruments: magnifier $1+D/f$; microscope $\frac{L}{f_o}\frac{D}{f_e}$ (small focal lengths); telescope $\frac{f_o}{f_e}$ (large objective) — and the mirror-objective arguments.

Morning-of-exam checklist: mirror plus, lens minus … mirror m has the minus, lens m doesn't … concave mirror f < 0, convex lens f > 0 … pool looks 3/4 deep … TIR needs BOTH conditions … at $D_m$ the ray parallels the base … powers add (dioptres!) … microscope wants small $f_o$, telescope wants large $f_o$ … reflecting telescope: chromatic-free, parabolic, back-supported. Go score.